. Proof, Let D be the input set of 2-intervals and let T (D), T ? and T ?? be the trapezoid families as described in the above description of Unl-{<

D. Also, {. Op-t-the-maximum, . ?}-comparable, D. Subset, |. T. We-have et al., Let ?(G T ?? ) denote the size of the maximal independent set of G T ?? . Since G T ?? is a forest

|. T. ??, Thus, the maximum independent set of G T ?? is at least of size 1 6 |OP T |, and the promised approximation factor holds, Accumulating all these inequalities together we get

A. Unl-{< and . ?}-approx, D) Data : A set of 2-intervals D. Result : A {<, ?}-comparable subset of D. begin 1, Construct T (D), the corresponding trapezoid set of D

T. Compute and . ??, a strongly proper subset of T ? , such that |T ??

G. Compute, T. ??, . Maximum, G. Set, and . ??, return the set of 2-intervals corresponding to the maximum independent set of G T ?? . end Fig. 6. A schematic description of algorithm Unl-{<

T. Complexity and . Let, Steps 1-2 in Unl-{<, ?}-Approx can be computed in O(n lg n) time by a similar analysis of the time complexity of {<

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